Calculus: Early Transcendentals, Chapter 6, 6.3, Section 6.3, Problem 26

The shell has the radius 5 - y , the cricumference is 2pi*(5 - y) and the height is 4 - x , hence, the volume can be evaluated, using the method of cylindrical shells, such that:
V = 2pi*int_(y_1)^(y_2) (5 - y)*(4 - x) dy
V = 2pi*int_(y_1)^(y_2) (5 - y)*(4 - sqrt(7+y^2)) dy
You need to find the endpoints, using the equation sqrt(7+y^2) = 4 => 7+y^2 = 16 => y^2=9 => y_1=-3, y_2=3
V = 2pi*int_(-3)^(3) (20 - 5sqrt(7+y^2) - 4y + y*sqrt(7+y^2)) dy
V = 2pi*(int_(-3)^(3) 20dy - 5int_(-3)^(3)sqrt(7+y^2) dy - 4int_(-3)^(3) ydy + int_(-3)^(3) y*sqrt(7+y^2) dy)
V = 2pi*(20y - (5/2)*sqrt(y^2+7) - (35/2)sinh^(-1) (y/sqrt7) - 2y + (2/3)sqrt((7+y^2)^3))|_(-3)^(3)
V = 2pi*(60 - 10- (35/2)sinh^(-1) (3/sqrt7) - 6 + (2/3)64 + 60 - 10 + (35/2)sinh^(-1) (-3/sqrt7) - 6 - 128/3)
V = 2pi*(88 - (35/2)sinh^(-1) (3/sqrt7) + (35/2)sinh^(-1) (-3/sqrt7))
Hence, evaluating the volume, using the method of cylindrical shells, yields V = 2pi*(88 - (35/2)sinh^(-1) (3/sqrt7) + (35/2)sinh^(-1) (-3/sqrt7)).